Hey euphony.
One question you need to ask is whether you want to include complex numbers as part of your range or co-domain of your function.
If you include complex numbers then things become a lot easier to define.
f(x) = x^x
So I think the domain would include:
a) all x where x > 0
b) all rational x where x < 0 and the denominator is odd (negative integers and fractions with odd denominators)
c) 0, if you define 0^0 to be 1 or 0. My two calculators give different answers for this.
But would it include negative irrational numbers? Would it include some and not others? Would there be a difference between algebraic and transcendental numbers?